Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2022-09-06
Simulation #1
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
2^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-3.53309 Min. :0.000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.67527 1st Qu.:0.627 1st Qu.:0.0000 1st Qu.:0.0000
## Median :-0.01628 Median :2.023 Median :0.0000 Median :1.0000
## Mean : 0.00034 Mean :2.137 Mean :0.4856 Mean :0.6036
## 3rd Qu.: 0.66792 3rd Qu.:3.447 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.82962 Max. :5.000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.485 2.796 2.746 4.188 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.9367 2.2983 2.3246 3.6511 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4553 1.7583 1.9538 3.1786 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.222 1.511 2.543 5.000
n = 200
results
## Initial lambda mean: 0.008063207## The selected undersmoothed lambda mean: 0.0001128359## The scaler picked: 0.01366887psi(a=0.5, z= 1)
psi(a=5, z= 1)
psi(a=0.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=5, z= 0)
Simulation #1_1
[Same as Simulation #1, but with posibility violation as the variance
of U_A decreased]
Data structure: \(O = (W, A, Z,
Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-4.039777 Min. :0.000 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.687173 1st Qu.:1.217 1st Qu.:0.000 1st Qu.:0.0000
## Median :-0.015235 Median :1.973 Median :0.000 Median :1.0000
## Mean :-0.006659 Mean :1.993 Mean :0.499 Mean :0.6582
## 3rd Qu.: 0.659240 3rd Qu.:2.731 3rd Qu.:1.000 3rd Qu.:1.0000
## Max. : 4.256538 Max. :5.000 Max. :1.000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 2.013 2.671 2.701 3.362 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.511 2.219 2.220 2.906 5.000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.060 1.746 1.770 2.436 5.000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.5197 1.1890 1.2479 1.8568 4.8720
n = 200
results
## Initial lambda mean: 0.01013305psi(a=0.5, z= 1)
psi(a=2.5, z= 1)
psi(a=5, z= 1)
psi(a=0.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=2.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=5, z= 0)
Simulation #1_2
[Same as Simulation #1, but with a 3-way interaction]
Data structure: \(O = (W, A, Z,
Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
2^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A + 0.3 * W * A * (Z+1) - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-4.039777 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.687173 1st Qu.:0.5447 1st Qu.:0.0000 1st Qu.:0.0000
## Median :-0.015235 Median :1.9332 Median :0.0000 Median :1.0000
## Mean :-0.006659 Mean :2.0698 Mean :0.4954 Mean :0.5825
## 3rd Qu.: 0.659240 3rd Qu.:3.3509 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 4.256538 Max. :5.0000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.304 2.587 2.591 3.951 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.8416 2.1806 2.2611 3.5800 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.3482 1.6997 1.9047 3.0925 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.129 1.473 2.480 5.000
n = 200
results
## Initial lambda mean: 0.007436044psi(a=0.5, z= 1)
psi(a=2.5, z= 1)
psi(a=5, z= 1)
psi(a=0.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=2.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=5, z= 0)
Simulation #1_3
[Same as Simulation #1, but with a 3-way interaction and positivity violations]
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A + 0.3 * W * A * (Z+1) - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-4.039777 Min. :0.000 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.687173 1st Qu.:1.217 1st Qu.:0.000 1st Qu.:0.0000
## Median :-0.015235 Median :1.973 Median :0.000 Median :1.0000
## Mean :-0.006659 Mean :1.993 Mean :0.499 Mean :0.6582
## 3rd Qu.: 0.659240 3rd Qu.:2.731 3rd Qu.:1.000 3rd Qu.:1.0000
## Max. : 4.256538 Max. :5.000 Max. :1.000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 2.013 2.671 2.701 3.362 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.511 2.219 2.220 2.906 5.000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.060 1.746 1.770 2.436 5.000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.5197 1.1890 1.2479 1.8568 4.8720
n = 200
results
## Initial lambda mean: 0.009272508psi(a=0.5, z= 1)
psi(a=2.5, z= 1)
psi(a=5, z= 1)
psi(a=0.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=2.5, z= 0)
Estimating ATE(0.5, 0) with z = 0
psi(a=5, z= 0)
